Aircraft: B767
Phase of flight: Climb
Anomaly: Deviation - Speed All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence

After departing Runway XX in ZZZ; initial climb up to 3;000 feet; we performed a low altitude level off and were in the process of cleaning the airplane up. As we went through 2;000 feet; we started to encounter moderate downdrafts and wind-shear in addition to the turbulence we had encountered since departure. In the process of performing the low altitude level off; the auto throttles began to retard; as the speed mode had not engaged. I disengaged the auto throttles and took manual control; the airspeed was uncontrollable and along with the downdrafts we received a 'don't sink' aural very briefly. I followed the flight director throughout the climb procedure and overrode the automation in an attempt to rectify the problem. We had communication issues contacting departure on the climbout; that along with the down drafts causing us to descend had ATC inquire if everything was okay. We responded yes; just correcting. After the momentary 'don't sink' the aircraft began to climb once again and we were able to climb out normally. The TAF showed WS for the area the entire day and day before of 45-55 kts. The CRJ ahead of us had not reported any adverse ride reports in our direction of flight. The wind shear and downdrafts were much worse than we had anticipated. The timing of the weather encounter being in the process of cleaning up as well as the auto throttles retarding led to an inability to climb; the problem was rectified by disengaging the auto throttles and pitching up to escape the weather.Hindsight being 20-20; I wouldn't have taken off knowing the situation; or waited for a ride report from the preceding aircraft. ATC advised us that center cleared our route; and we took that into consideration in our decision to go. ATC had offered us depart ZZZ at up to 12;000 feet in order to get past an approaching front. A CRJ took off a few minutes before us and did not report any issues and another regional jet landed as we held short of the runway with that aircraft on a 6 mile final. When we were cleared for takeoff; the First Officer read back the takeoff clearance and I taxied the aircraft onto the runway and gave the controls to the First Officer (now Pilot Flying--PF) for takeoff. We were cleared to takeoff and told to fly runway heading to 3;000 feet. Prior to even taking the runway; we discussed the likelihood of a low-altitude level off and what to do. The takeoff roll was uneventful although I noticed some variation in airspeed which I attributed to wind gust and I advised the First Officer to delay rotation to get a little more speed for a better margin of safety. We rotated and started the climb out; which quickly became bumpy with continuous moderate turbulence. Although turbulent; the profile and acceleration appeared normal. At around 1;000 feet AGL; the airspeed seemed to be approaching the max flap speed (we were at flaps 15) and we began retracting flaps. At approximately 1;800 feet; the PF called for 'Speed; set Speed'. I promptly did this and dialed up the speed toward approximately 220 KTS; however the thrust levers began to retard more than necessary. At the same time; I attempted to make my first contact with ZZZ Departure. The PF mentioned the thrust levers were coming back too far and pushed them forward to countermand them. I then noticed the speed reference indicator had reduced to somewhere down near V2. I quickly dialed the speed in the speed window back up to 220; however as I looked at the instruments; I saw that the airplane had lost altitude (approximately 200-400 feet). At no time had the PF commanded the airplane attitude below the flight director but I advised him to come back up. We had already started correcting when we heard the aural 'Don't Sink.' I made a second attempt to contact ATC as I heard the controller repeat himself. I looked and noticed my audio selector was misconfigured. I quickly corrected and contacted ATC. The Controller asked if we were OK and I told him we were fine. He said he was checking on us because he had not heard us check on and saw that we had descended. I advised him that we had corrected that and continued climb. He then cleared us to a higher altitude. The weather continued to be an issue and we quickly had to deviate to avoid several thunderstorm cells painted by radar. The radar was painting light to moderate precipitation so I turned the engine ignition to CONT for the duration of the climb. We were able to proceed on to ZZZ1 without incident. Although we did not receive any warnings of windshear; this may have been a windshear / downdraft event given the descent encountered. Although we quickly corrected for the airspeed and sink; the performance of the aircraft and the actions of the crew should have resulted in a steady climb; even though conditions became turbulent. From this experience I would recommend crews in a similar situation query ATC for a PIREP from a recently departed aircraft; such as the CRJ which took off a few minutes before us. I also should have verified my audio panel configuration prior to take-off to avoid the communication confusion. B767 flight crew reported encountering strong wind shear and moderate turbulence during climbout which triggered an aural warning.